Download - IBK - 423 NUCLEAR ENGINEERING
IBK - 423
NUCLEAR ENGINEERING
IBK-423 S.Krčevinae and S.Iakač
MEASUREMENT OF THE EPEECT OF THE LATTICE PITCH ON THE
EPPECTIVE RESONANCE INTEGRAL OF NATURAL URANIUM
BORIS KIDRIC INSTITUTE OF NUCLEAR SCIENCES BEOGRAD - VINCA
April 1966
MEASUREMENT QE THE EFFECT OF THE LATTICE PITCH ON THE EFFECTIVE RESONANCE INTEGRAL OF NATURAL URANIUM
by
S.Krčevinae and S.Takač
INTRODUCTION
The analytical theory of resonance absorption, as well as the numerical Monte Karlo method, allows calculation of the resonance integral. However, it is based on specific approximations so it may be used accurately enough in a limited number of cases« Likewise, insufficiently accurate knowledge of the basic nuclear constants (e.g. resonance parameters, etc.) used as input data in analytical calculation, leads to inaccurate determination of the resonance integral.
Therefore, experimental determination of the effective resonance integral is still indispensable. In some cases the experimental results are used as the exclusive source of information, in other as "the basis for the semi-empirical technique of calculation, and in others as a check of the new theoretical procedures.
There are several experimental methods of direct determination of the resonance integral; the activation method (1, 2), the reactor oscillator and the danger coefficient method. Indirectly, using the results of critical
- 2 -
experiments, it is possible to determine correlated values
of the effective resonance integral.
The present work investigates the dependence of
the effective resonance integral on the lattice pitch,
Theoretically, the dependence is determined starting from
Wigner's rational approximation in which the lattice is
characterized by the effective ratio (S/1T) . later this al
lows correlation between the lattice resonance integral and
the resonance integral of the single rod (the rod in infinite
medium). Using two approximations for Dancoff 's factor we
give the measured functional dependence of the effective
resonance integral on the effective (S/M) ratio.
To determine the resonance integral experimentally
we used the activation method and the differential texhniq^e pio
of measuring absorption distribution in U , Since, because
of the use of cadmium in determining the ca,dmium ratio in
the fuel rod, the effective lattice pitch cannot be defined
with certain reliability, besides other perturbing effects,
the method of comparing thermal activations of U and a
suitable thermal detector are used to determine the cadmium
ratio.
- 3 -
1. THEOEY OF EXPERIMENT
Thermal reactors have an epithermal flux which
changes with energy approximately like l/E. For such flux
distribution the resonance integral is defined in the fol
lowing way;
jr t> -W f
1.
where
F(E,r) - flux depression factor in the rod
f(E,T,A) •- factor assuming Doppler's effect
V - absorber volume
In ihtegral(l) E is the lower limit of the epithermal flux
and in the case of a thermal reactor it may he taken as 5kT
where T is the temperature characterizing Maxwellian compo
nent of the neutron flux or, as usual in practice, as the
energy of the effective cadmium cut-off. In the case of 238
absorbers such as U with resonances at energies fairly
higher than thermal, exact definition of the lower limit is
not of essential importance. Since the integrand in (1)
rapidly decreases with increasing energy, the integral con
verges and the infinite upper limit is conventionally taken.
If the neutron flux below E is denoted by 'f, ^ , O f .•')
and that above E by 0^^;, the neutron absorption rate in
the given bare and cadmium covered material is expressed
- 4 -
by A / s
A6 ~ NV^&th + M\/fI?I)^ 0 ^ / ; /
Ac^ = A / i / f ^ / y ^ f ^
where fpt^ and ^ w đo not depend on energy,
Defining the r e l a t i o n by A h
we obtain
^GaL
•-h & _-*
— — — ~ ~ ~ ~ ~ ~ ~ f '•- * • • * . , / •
The relation allows determination of HI provided
the thermal to epithermal flux ratio in the reactor is known.
By measuring the cadmium ratio for a material of a known
resonance integral it is possible to determine this ratio
and obtain the relation
,' /V" l ~ — — - 11 ( h-J_ / 1
• "•CH'V « 0 -'-'-
where J
X - material of unknown RI
st - material of known EI
60 - effective cross-section for absorption 2200 m/s
(HI), + thus defined also includes l/v component of the
effective cross-section*
Hie above relation is derived under the following
assumptions;
1, The epithermal flux varies approximately like l/E
_ 5 -
2. The unknown sample and the standard have an
absorption cross-section which is changed
according to the l/v law in the thermal region,
3« Self-protection for thermal neutrons is
neglectable.
4. for the standard the self-shielding for reso
nance neutrons is also neglectable.
In the given case assumption 1. is implicitly
adopted. Since deviation from the l/v law of absorption in 2^8 197
U and Au is small, we considered assumption 2.to be
also fulfilled for the given materials. In the case of
Au-Al-alloyed foils, assumptions 3. and 4. were also fulfilled,,
whereas in the ease of pure gold foils corresponding cor
rections were made. Gold was chosen as the standard because
it has the most precisely determined resonance integral.For
the uranium rod assumption 3.was not fulfilled so the cor
rection was experimentally determined.
In measuring H„ for the fuel rod, the use of
cadmium induces considerable perturbing effects which cannot
be easily treated theoretically and the experimental cor
rections would be numerous. Hence we chose the method of
measuring the cadmium ratio which is based on the comparison 238
of thermal absorptions in U with thermal absorptions in a suitable thermal detector /5, &/.
The method consists in the following:
According, to the definition the cadmium* ratio of
the fuel rod is:
- 6 -
where s
r - Irradiation in the lattice in which the cadmium
ratio of the fuel rod is determined
1c - irradiation in pure thermal neutron flux, in our
case in the thermal column of the Vinca M
reactor
U - fuel element, natural uranium rod
Dy - disproslum foil.
It is assumed that the total neutron flux may be
divided into the thermal component .£ + ! and the epithermal
component &--£,;. The respective activities are denoted by
A,, and A .. It is also assumed that 0n_ « 56 ., and th epi Od * epi A_n K A ., i.e., the lower limit of the epithermal flux is Od epi the energy of the effective cadmium cut-off*
With the adopted assumptions it follows that:
A" <* - ,b ' f/
^ , , -1 l a '
e: -1 •CvL
h '
™
—
v» /<* -
* th * ,pi
*':• th
^
It is seen from relation (4) that if we want to determine U U tr H» , besides activity A, , we should also toiow A,,« To determine A., we directly use a suitable thermal detector,
164 i.e. in our case Dy .. lor disprosium we analogously have.
- 7 -
I f we d i v i d e ep i l a t ions (4) and (5) We obtain.:
/ * \ -1 '/
/
My ~s?
i AJ: I! 'A** / f e^y
-<*/ l * /rl --tH/r [ The relation of the saturation activities of uranium and
etisprosium is expressed "by:
A fy} ! /v fy ,, '1/ ,~.' %-, k.t '' I 7,
By irradiating the same samples in a well-thermalized flux
we obtains
/' , U \ / It ' •* /'
•fit
By dividing (7) by (8) we obtain;
/
8.
9.
Since:
4 / 4
For equation (6) we finally haves
10.
6' u.
- a -
def in ing Gs
\ *t<PttJr \ 'J"&H JK 12'
u As may be seen from the above relation B„. ean be
-v_ TT Cd
determined only "by knowing R„"i j A_, need not "be known« Since
Dy, which has a large Rp, in the lattice, was the reference
detector, the error induced by cadmium was neglectable. On
the other hand, since disprosium has a cross-section which
does not vary quite according to the l/v law, factor & is
to some degree unreliable. In case the reference detector were a pure l/v detector, factor £ would he equal to unit. Relation (11) shows that (!„,)k should also he determined in
vd.
the case of irradiations in well-thermalized flux. However,
since the flux is well-thermalized the cadmium ratio is very
high so the corresponding factors in relation (11) are very
close to unit and the error with which they are determined
-slightly affects the result- Factors related to irradiations
in well-thermalized flux are absolutely independent of ir
radiations in the lattice so that once determined they may
he used for all experiments.
Considering the relation defining factor G- it may
he taken into account that Dy is not subject to the l/v law
and the neutron temperature in the lattice and the Well-
thermalized flux is not the same. This factor may be theo
retically determined ass
^ ~~ "J* &c, (t)0 ft }d£ '-" « * ' • • » ' • .. '• '. . ' i . im — « • • • • • . , , , - . — , , » . . » . i
- 9 -
Using the fact that factor & slightly depends on
the temperature, it is taken from reference (5) with as
sumption that the iaeutron temperature is 340 E. Such an
error is considered neglectably small compared with the 238
error of the result. In factor G-? U is considered as the
1/V absorber.
Thus are obtained values for the resonance integrals
for different lattice pitches. This dependence is interpreted
by using two approximations for Dancoff's factor in Wigner's
rational approximation. It is known that in this approximation
the effective resonance integral of the uranium rod in the
fuel element lattice is correlated with the resonance in
tegral of the single rod. This correlation is based on the
effective (S/K) ratio of the rod in the lattice.
In Wigner's approximation we gave the well-known
relation for this ratio
{ ijrj = {-ŽT <'>' V
where (1-C) is Uancoff 's factor and S/ffl. the surface to mess
ratio of the single rod. The accuracy of determining this
factor depends on the fuel geometry and the configuration
of the fuel element. Using Bell's approximation Dancoff's
factor is expressed by
r 4
i d. > 1/ i
Where
S-./Y-, - moderator surface to moderator volume ratio
y - macroscopic cross-section for moderator
scattering.
- 10 -
2. EXPERIMENTAL PROCEDURE
2.1. Detectors
238
To determine the absorption distribution in U ,
foils of 4-0$ alloyed slightly depleted uranium and aluminium,
10 mm in diameter and 0.04 mm thick, were used. Mass spectro-
metric analysis showed that the foils contained 0.220-0.003$ 235
U * The foils were intercalibrated for their natural
activity. The statistical error was 0.7$. The thermal neutron
distribution was measured with Dy foils of 5$ alloyed dis-
prosium and aluminium, 4 mm in diameter and 0.14 mm thick.
These foils were intercalibrated by rotation of all foils in
the central pit of the RB reactor. This made possible ir
radiation of all foils by the same flux. The induced /3 -
activity of the foils was measured with the same set of GM
counters. The results obtained were calculated on a computer
with standard corrections to decay and dead time. The foils
were intercalibrated several times so the statistical error
of the factors is less than 0.1$,
We have mentioned above that gold was used as
standard. Two kinds of foils were used; one was of 0,05$
alloyed gold and aluminium, 10 mm in diameter and 0.1 mm
thick, and the other of pure gold, 10 mm in diameter and 20
microns thick. The former had neglectable self shielding and
they were intercalibrated in the thermal column of the RA
reactor. The statistical error was 0.5$-. This error was
reduced by the choice of foils with approximately the same
calibration factor. The gold foilg were intercalibrated ~by
measuring their weight on a precise balance. They had
- 11 -
considerable self; shielding so that the cadmium ratio had to
"be corrected using data from the literature /4/. Results of
measurements for "both types of foils were in good agreement
within experimental errors. During measurement, the gold
foils were coated With 0.76 mm thick cadmium cover.
2.2. Description of the experimental technique
The resonance integral was measured in the natural
uranium - heavy water system. The fuel element had the form
of a rod, 25 mm in diameter, coated with 1 mm thick aluminium.
The absorption distribution was measured in the
fuel element by the differential technique, characterised
with following advantagess
- streaming of epithermal and thermal neutrons is
negligible
- the surface effect in resonance absorption can
be determined with better accuracy
- suitable foil-counter geometry
- space distribution of both thermal and 238
epithermal absorption in U is obtained.
The disadvantages of the technique such as: the
large number of measuring foils and reduced statistics of
activity detection may be easily avoided by well-organized
experiment.
238 The absorption distribution of U was measured on
..: a specially prepared fuel element with a radial hole
11 mm in diameter* In a specially prepared copper tube with
0.45 mm thick walls, ten D-Al foils were positioned along
- 12 -
the diameter of the rod as is shown in 3?ig.(l). Each foil
was placed between two 0.05 mm. thick aluminium catcher
foils. This was done so as to avoid deposition of the fis
sion products from natural uranium spacers placed between
the foils. The diameter of the spacers was 10 mm and they
had different thicknesses. Thicknesses less than 1 mm were
machined on a lathe and spacers thinner than 0.05 mm were
rolled on a special roller with high rolling accuracy. To
obtain as best detection statistics as possible, 10 mm
đia U-Al foils were used. Consequently, this Was dangerous
because the foils near the surface of the rod could be
wrongly positioned and their proper position would not be
known for certain because their surface would intersect a
large number of equiflux lines. To avoid wrong positioning
of the foils on the fuel surface the spacer placed at about
10 mm from the centre of the rod was specially treated so
that one of its surfaces was flat and the other had such a
radium that the foils arranged towards the rod surface were
slightly bent so that the surface foil exactly followed the
surface of the rod. To obtain highest possible accuracy and
determine the possible asymmetry in positioning, the foils
on the surface and next to the surface had their symmetric
foils arranged as is shown in Jfig.l.
The thermal neutron distribution was measured
with Dy-Al foils by the extrapolation technique described
in details in paper (7). In our case one Dy-Al foil placed
on the surface of the aluminium coating was simultaneously
irradiated with the U-Al foils. Using the thermal distri
bution from paper (7) we obtained integral activity.
Together with the above foils we irradiated bare
- 13 -
and cadmium-covered gold foils which were also on the surface
of the aluminium coating and were therefore irradiated by
the incoming resonance flux. Care was taken that hare and
cadmium covered foils would be in the same flux. The gold
foils were at least 30 cm far from the hole in which the
U-Al foils were placed so that possible perturbation induced
by cadmium would be avoided.
The measurements were made for the single rod,i.e.
for the rod placed in the centre of the thermal pit 40 cm in
diameter and in the system of 7,8, and 11.3 cm lattice
pitches*
2.3. Sample measurements
The activity of all the foils was measured on a
set of counters shown in Pig.2 consisting of four Philipps,
model 18505, &M counters. To reduce their relatively high
dead time, we applied electronic quenching using Zener
BZ-450 diodes. The electric circuit used is shown in Pig.2.
The GM counters were connected with two sets each consider
ing of four decade scalers. Each set had two timers Which
automatically controlled the time of measurement and pause.
This considerably facilitated the measurement and improved
the accuracy.
The f6 -activity decay of the U-Al foils was fol
lowed usually after 30 minutes after the completion of ir
radiation. Care was taken that each foil would be measured
on the same counter on which it had been calibrated so as to
avoid the necessary interealibration of the counters. To
obtain, as best statistics as possible the central and the
- 14 -
surface foils were measured permanently- on two counters
while the rest of the foils were measured on the other two 239
counters alternately in cycles. The decay of U was usual-239
ly followed for 10 periods and then the decay of Np was
followed during the next week when the activity fell on the
"background. 239
The activities did not originate only from U 239 -
and Mp which is proved "by the fact that they did not
follow the decay corresponding to the cited isotopes. The
undesired activity originated from fission products "because
the U-Al foils were slightly depleted and they contained 235
0,22$ U . Therefore, the cited activities had to he corrected.
2.4. Calibration irradiation in the thermal flux
The calibration factor which allows comparison 238 164
of the thermal activations in U and Dy was first
determined in the thermal pit of the HB reactor by rotation
of the bare and cadmium covered U-Al and Dy-Al foils. 238
However, since the cadmium ratio for U was not high
enough, to obtain better accuracy we irradiated the same
foils in the thermal column of the RA reactor.
2.5. Correction to fission product decay
The undesired activity of the fission products
from insufficiently depleted foils made impossible accurate U determination of &«•,« In order that the measured activity
would be corrected in some way we used the following pro
cedure.
- 15 -
In the time interval from 30 - 230 minutes after
the completion of irradiation the activity of the foils.
decay according to the laws
A,^A„e •%. AH e h(j + A..+ + B i s . .' ' 1 •' rip t "
and after 230 minutes according to the laws
A*r*-A-Ht>- p'-hAfi ~+8 13
In the above two relations the symbols denote:
A ~ measured activity in the time interval from m 30 - 230 minutes.
A*'"'- measured activity after 230 minutes m 239 A - initial activity of U U 239
kKi - decay constant of U 239 A.. - initial activity of Np
\ 239
A14 - decay constant of Np
A«, - initial activity of fission products
B - foil "background plus counter "background In the above decay laws the fission products are
-1. 2 assumed to decay according to the law t ' , According to equations (12) and (13) a programme
was made for the ZTJSl Z-23 computer which consisted in the
following. The value for A~ was graphically guessed from p (2)
the measured activities of A . It was used as the initial m
input value in equation (12) where the activity of A„
iteratively varied until the criterion that the obtained
activity satisfies the decay constant A u is fulfilled. Such
value for A_ was introduced in equation (13) where iteration
- 16 -
of B was allowed until the value obtained satisfied the
decay constant /vN , The obtained values for JL, and B were
again introduced in (12) where A„ iterated again until the
criterion required is satisfied. From this point the whole
procedure was repeated and iterations lasted until both
criteria were satisfied.
As output of the programme we obtained the required
activities of A., and A- , as well as A„ and B. An example of
such results is given in Table I,
Besides corrections to fission products the pro
gramme made corrections to radioactive decay and dead time
as well. It should be noted that equation (12) is not 233
exactly valid, because Np is the daughter product of 239
U , but we think that the possible error which is intro-239
duced owing to this fact is small, because Np has a long 239
half-life compared with U .
- 17 -
3. EXPERIMENTAL RESULTS
To find the mean value of the fuel rod activity
on the basis of the results of the programme, graphical
integration was performed. The distributions normalized to
the centre of the fuel and the calculated mean fluxes are
shown in Table II. The measured distributions are also shown
in Fig«3. It is obvious that the gradient of the measured
distribution very quickly increases with decreasing lattice
pitch. This is natural with respect to the increase of ab
sorption on the rod surface with relative increase of the
epithermal component in relation to the thermal component
by decreasing the lattice pitchy Since the distributions
have a rather low gradient starting from the centre of the
rod, the use of a large number of foils near the centre was
not necessary neither was the requirement for exact position
ing so strict as it was for foils near the surface. According
to our estimation the positioning error was 0,01 mm taking
into account the tolerances with which the U-spacers and the
experimental arrangement were made. This error could have
caused the value of the flux on the surface to be inaccurate
even up to 10$ in the case of the densest lattices. However,
this error has very little influence on the mean flux because
of the very sharp gradient near the surface of the fuel.
Correction of the saturation activities due to the
presence of fission products was even up to 25$ depending on
the place of the foils. This induced the greatest uncertadn+y
in the result. Because of the low statistics of the measure-239
ment of Bp decay the values for saturation activities
had a statistical error even up to 7$. This error could be
- 18 -
avoided only "by increasing the reactor power or the ir
radiation time. However, for the sake of security of
operation of the KB reactor, the irradiation was limited
to maximum 20 Wh. If it is limited only to the measurement 239
of U it is hetter to take the shortest possible time at
the highest possible power.
For the ahove reasons the mean fluxes and therefore U U
the activities of (A.) used in the formula for R„ have an
error of 3—4$ in dependence on the lattice pitch, Due to
the high statistics of measurement the activities of A, * U
have an error of 0*1 $, The relative error of B„, measured Cd
in the lattice is determined "by the expression £ X
%£*." 1~K
where:
I A ' - ' -•'
/ ' /P^ y _. 1 \
v Cd / r
Ab/
£ ., .' .£:. i / /j/y I
't>/r ! 4/ A..
- 19 -
A c -A
The error in factor G- is determined on the basis of work (5)
and its maximum is 0.5$.
The error of measuring the resonance integral was
calculated using the expressions
It may he seen that the error of determining the
nuclear constants was not taken into account.
Au The error in determining H-,., using two kinds of
Cd u • . .
gold foils was 1$.
The cadmium ratios as a function of the lattice
pitch and the resonance integrals obtained are shown in
Table III« (HI), , denotes the total resonance integral and tot
RI the resonance integral in which the l/v component is sub
tracted. Since the cadmium ratio for the uranium rod is
determined without the use of cadmium, the l/v contribution
is taken to he 1,1 b as usual. The values with which R„, is determined are given
Cd
in Table IV. The resonance integral as a function of the
lattice pitch is shown in Pig.4« The effective resonance
integral of the single rod is correlated with the lattice
resonance integral using two expressions for Dancoff 's
factor. Numerical data from ANL - 5800 and well-known
Bell's expression were used. A plot of the effective
resonance integral against the effective (B/M.) ratio
is shown in Pig.5.
- 20 -
4. DISCUSSION
Measurement of the resonance integral is one of
the most complicated intracellular measurements, The effect
of perturbation cannot he avoided at all and the accuracy
of the corrections is always unreliable. Besides, the
relatively great number of corrections increases the un
certainty of the results. In our case the greatest uncertainty
in the results was induced by the detector of absorption in 238
U because the interference activity of fission products
was relatively high. The correction made by analysing the
complex decay are considered correct with the following
remarks. In the most suitable cases the corrections were
even up to 25$ which is considered rather high. Furthermore, -1 2
in analysing the complex decay the t ' law was used. To
check this law we measured the decay of fission products
using aluminium catchers. The purpose of the experiment was
first to determine the time the activity takes to fall to 239
minimum so as to determine the most suitable time for Kp
measurement. The decay obtained did not correspond to the -1 2 t ' law. This was the reason for suspicion of the worthir.^oQ
-1 2 of using the t ' law in our case. We consider that special
attention should be paid to this in future, because, by
using natural uranium foils and by accurate analysis of the
complex decay it would be possible to separate the group of
interfering activities. To prove the advantages of the dif
ferential over the integral technique we made integral
measurements* The results obtained are constantly considerably
higher than those obtained by the integral technique,However,
since the results were badly reproducible we cannot give
- 21 -
them a definite weight.
The method of comparing thermal activities in U determining B.„ has shown, besides the above advantages,
that particular attention should be paid to the determination of the calibration factor by performing the irradiation in the thermal column. Since irradiation in complete thermal flux is concerned, the activity due to thermal absorption is comparable with the interfering activity of fission products, Therefore, the corrections here are the highest so they cause most of the errors of the results« The statistics with which this factor is determined should also be increased.
By using completely depleted foils and measuring 239 239 the gamma activities in U and Fp , the correction would
be completely reduced so the uncertainty of the results would be considerably reduced too.
To investigate the variation of the resonance spectrum as a function of the lattice pitch, i.e. deviation from asymptotic beliaviour of l/E, special measurements of the neutron flux spectrum should be made in future work and the resonance integrals correlated with this variation.
ACKNOWLEDGMENT
We would like to express our thanks to Er.Nenad Eaišić for the useful discussions and valuable suggestions. We also thank B.Mitrović for her assistance in measurements and the staff of the EB reactor anđ R.Hajđarevie for their cooperation.
REFERENCES
1. E.Hellstrand; "Measurements of Effective Resonance Integral in Uranium Metal and Oxide in Different G-eometries", Journal of Appl.Phys. Vol.28, No.12, 1957.
2. E.Hellstrand: "Experimental Studies of Resonance Integrals" in "Heavy Water lattices" IAEA Panel, 1963.
238 3. J.Hardy et al.: "Effective U Resonance Capture Integrals
in Rods and Lattices" in "Naval Reactor Physics Handbook" p.1276.
4* G.Jacks: "A Study of Thermal and Resonance Neutron Elux Detectors" DP-608.
5. S.Larvin et al.: "Methods for Determination of f*2'- and <54c Based on Chemical Separation of Np 239
and M Q 5 9 from Uranium and Eission Products" KIR - N 26.
rv(t
6. R.Lewis et al.: "Thermal Activation Method for f'' Measurements in Slightly Enriched Uranium Oxide Lattices", BAW - 1268.
7. S.Takač and S.KrSevinac: "Extrapolation Method for Measurements of Neutron Elux Distribution in Reactor Cell" - to be published.
TABLE I
Calculation of the complex decay
Foil N2 >i
it
23 23 23
23 23 23
23 23 23
Z m
B
B
B
147.0 151.8 151.8
151.8 151.8 151.8
151.8 151.8 151.8
= A1 (Exp)
A3 A3 A 3
11.7 11.7 11.7
13.9 13.9 13.9
14.06 14.01 14.01
(-I t) +
A l A2 Al
12054.3 168.4 10002.5
11638.0 166.0 11372.5
11550.7 165.8 11533.8
A2(Exp) (-l2t)
SA1 SA2
SA1 0.83/-01 0.98/-02 0.41/-01
0.57/-01 0.97/-02 0.80/-01
0.86/-01 0.97/-02 0.86/-01
+ At""1'2 +
Ll L2 Ll
0.29/-01 O.20/-03 0.26/-01
0.29/-01 0.20/-03 0.29/-01
-0.29/-01 -0.20/-03 -0.29/-01
B
SI
SL2
SL1 0.65/-05 0.18/-09 0.32/-05
0.69/-05 0.17/-09 0.63/-05
0.68/-05 0.17/-09 0.67/-05
TABLE I I
l / r o
7 cm 1.0000
8 cm 1.0000
1 1 . 3 eta 1.0000
oo 1.0000
0 .5 looo
1.1608 1.4366
1.1645 1.3827
1.1585 1.2351
1.1600 1.2100
l . o 6 7 1.191
1.4606 2 .5631
1.5603 2 .5171
1.3546 1.7520
1.2478 1.2801
l,21o 1.229
3.5547 5.2874
2.8556 3.6112
2.3570 3.7467
1.3400 1.4997
1.250 0
9.5137 1.7856
7.7602 1.6862
4.6158 1.5238
1.9003 1.2476
TABLE I I I
i G
7 cm 1.021
8 om. 1 . 0 1 5
1 1 . 3 cm 1.009
oo 1.001
RK *
0.998 0 .960
0 .998 0 .971
0 .998 0 .588
0 .988 0.995
62.55 0.0100
68.45 0.0100
81.02 0.0100
95.4-8 0.0100
1.1747 1.4652
1.1861 1.4750
1.2004 1.5060
1.2115 1.510
TABLE IV
1
7 cm
8 cm
11.3 cm.
CD
U RCd
2.58 i 3f*
3.053- 3$
5.150- 5f°
20.48 - 4.596
RCd
1.390 i 196
1.521 i 1$
2.089 - 1 . +
6.20 - 2 .
5S* 5f0
(BI), . t o t
10.70 i 8.4^
10.97 i 7.5f«
11.32 - 7.2fo
11.53 - 7 .3^
EI
9.60 i 8.4?o
9.87 - 7.5$
10.22 i 7.2^
10.43 i 7-3^
FIG. J
uranski otsojhik
U-Al. foiJ/e
Al. hvatae
2
t i
1 OS
dwo
1 1 Lws?
0M+
« /=oo -
1 — * R (cm)
h250
-1.229
1210-
FIG. 2
GM
HV, r
C iS "**"» SK- IJ
n r * RELCY
* i
TIM CD 1 IV
SCALER
47M
XBZ4S0
If WOpF
to scaler
82K
+HV
mm
it -
10
KImmjm^w-43b
• FIG. 4
10 15 20 l (cm)
mm
11 -
10
o Belt-approximation
* Numerical calculation, ANL~ 5800
FIG, 5
0.27 0.28 0.29 030 ]/(ŠŽMf(cm